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clarified, removed useless definition of equality and added needed definition of minimum
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YCor
YCor
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Let $G$ be a finite group and let $n$ be the order of the groupsorder $G$$n$. Two

A generating sets areset in $G$ is said to be differentminimum if at least one element in the two generating set is differentit has minimal size. 

Is there a known lower bound on number of minimum generating sets in a group of order $n$? For cyclic groups I know the answer.

Let $G$ be a finite group and let $n$ be the order of the groups $G$. Two generating sets are said to be different if at least one element in the two generating set is different. Is there a known lower bound on number of minimum generating sets in a group? For cyclic groups I know the answer.

Let $G$ be a finite group of order $n$.

A generating set in $G$ is said to be minimum if it has minimal size. 

Is there a known lower bound on number of minimum generating sets in a group of order $n$? For cyclic groups I know the answer.

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user108347
user108347

How many minimum generating sets are there in a finite group?

Let $G$ be a finite group and let $n$ be the order of the groups $G$. Two generating sets are said to be different if at least one element in the two generating set is different. Is there a known lower bound on number of minimum generating sets in a group? For cyclic groups I know the answer.